The offline planning of Cartesian trajectories of three-degrees-of-freedom (3-DOF) spherical wrists is considered. Motions undergone by such wrists are regarded as points on the surface of the unit sphere centered at the origin of the four-dimensional space of the linear invariants of the rotations involved. The projection of the sphere onto the three-dimensional space of the linear vector invariant of the rotation sensor is shown to be a solid unit sphere centered at the origin of the three-dimensional space. Thus, a rigid-body rotation given as a smooth function of time appears as a smooth curve within the unit sphere of this space. A one-to-one relation between the time derivative of this 3-D vector and the angular-velocity vector yields the kinematic interpretation of the velocity of the point tracing that curve. This concept is applied to a problem of trajectory planning for robotics pick-and-place operations using a spline-based concept of curve synthesis introduced elsewhere.>